Question

How do I intuitive figure out if a question needs the use of Integration or Differentiation?

Answer 1

I would have to disagree with this premise. Integration is antidifferentiation. Every rule you have for the one is echoed in the other. The derivative of a product becomes integration by parts. The chain rule becomes u-substitution. The derivative of a function inverts to become the integral of the derivative. Finally, in the realm of numerics, it is the exact opposite — numerical integration is far simpler, and much more stable, than numerical differentiation because of the loss of precision when subtracting two large numbers to form a small number (and then dividing it by another, similarly formed, small number). The only reason the differentiation of functions succeeds so well in a classroom is because the classroom confines the study of derivatives (and integrals) to the comparatively small set of functions where the rule (which always functions both ways!) can be derived in a closed, understandable form.

A better question (addressed elsewhere on Quora) is why differentiation is taught first. A second useful observation is that Newtoninvented differentiation in order to formulate physics, and most useful calculus is formulated even now in terms of differential equations first, which are then integrated to find their solutions. Curiously, in almost any context where one actually uses calculus, the actual solutionprocess is integration, while the problem itself is formulated in terms of derivatives. One actually spends much less time differentiating than one does integrating (or solving differential equations, which is in some sense integration on steroids).